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Properties

Label 2-175-5.2-c4-0-0
Degree $2$
Conductor $175$
Sign $-0.973 + 0.229i$
Analytic cond. $18.0897$
Root an. cond. $4.25320$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0151 + 0.0151i)2-s + (−8.40 + 8.40i)3-s − 15.9i·4-s − 0.255·6-s + (13.0 + 13.0i)7-s + (0.486 − 0.486i)8-s − 60.2i·9-s + 178.·11-s + (134. + 134. i)12-s + (−130. + 130. i)13-s + 0.398i·14-s − 255.·16-s + (−219. − 219. i)17-s + (0.915 − 0.915i)18-s + 255. i·19-s + ⋯
L(s)  = 1  + (0.00379 + 0.00379i)2-s + (−0.933 + 0.933i)3-s − 0.999i·4-s − 0.00709·6-s + (0.267 + 0.267i)7-s + (0.00759 − 0.00759i)8-s − 0.743i·9-s + 1.47·11-s + (0.933 + 0.933i)12-s + (−0.770 + 0.770i)13-s + 0.00203i·14-s − 0.999·16-s + (−0.758 − 0.758i)17-s + (0.00282 − 0.00282i)18-s + 0.706i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(18.0897\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.07839023002\)
\(L(\frac12)\) \(\approx\) \(0.07839023002\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-13.0 - 13.0i)T \)
good2 \( 1 + (-0.0151 - 0.0151i)T + 16iT^{2} \)
3 \( 1 + (8.40 - 8.40i)T - 81iT^{2} \)
11 \( 1 - 178.T + 1.46e4T^{2} \)
13 \( 1 + (130. - 130. i)T - 2.85e4iT^{2} \)
17 \( 1 + (219. + 219. i)T + 8.35e4iT^{2} \)
19 \( 1 - 255. iT - 1.30e5T^{2} \)
23 \( 1 + (-54.1 + 54.1i)T - 2.79e5iT^{2} \)
29 \( 1 + 438. iT - 7.07e5T^{2} \)
31 \( 1 + 1.44e3T + 9.23e5T^{2} \)
37 \( 1 + (914. + 914. i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.73e3T + 2.82e6T^{2} \)
43 \( 1 + (-103. + 103. i)T - 3.41e6iT^{2} \)
47 \( 1 + (1.42e3 + 1.42e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (3.30e3 - 3.30e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 4.66e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.97e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.72e3 - 3.72e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.24e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.43e3 + 3.43e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 1.11e4iT - 3.89e7T^{2} \)
83 \( 1 + (3.63e3 - 3.63e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 8.15e3iT - 6.27e7T^{2} \)
97 \( 1 + (4.38e3 + 4.38e3i)T + 8.85e7iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.08084447442135962246222415798, −11.45787881645817196654083956518, −10.64422777113667818391108754879, −9.601343159607548646312674152200, −9.049420164941599232544661311645, −7.01288662747388276044730786210, −6.00776433899461337135901470064, −5.00643865069478135497115364913, −4.14313838410593854408169505693, −1.77781504069810076375905562212, 0.03230699502263248476399893919, 1.68686732016248887455251812765, 3.54517669029203878701394466731, 4.96488599112733218660189380869, 6.51625665422745076302251578542, 7.08436755325643515329304326611, 8.199585631205806999692154867817, 9.339089879253611489392434381593, 10.98936507496006216729179906526, 11.61041223538823319360415468733

Graph of the $Z$-function along the critical line